Electronic Journal of Qualitative Theory of Differential Equations 2012, No.17, 1-13;http://www.math.u-szeged.hu/ejqtde/
POSITIVE SOLUTIONS OF
NONLINEAR FRACTIONAL BOUNDARY VALUE PROBLEMS WITH DIRICHLET BOUNDARY CONDITIONS
QINGKAI KONG† AND MIN WANG‡
Abstract. In this paper, we study the existence and multiplicity of positive solutions of a class of nonlinear fractional boundary value problems with Dirichlet boundary conditions. By applying the fixed point theory on cones we establish a series of criteria for the existence of one, two, any arbitrary finite number, and an infinite number of positive solutions. A criterion for the nonexistence of positive solutions is also derived. Several examples are given for demonstration.
1. Introduction
In this paper, we study the boundary value problem (BVP) consisting of the fractional differential equation
−Dα0+u=w(t)f(u), 0< t <1, (1.1) and the Dirichlet boundary condition (BC)
u(0) = 0, u(1) = 0, (1.2)
where 1 < α < 2, w ∈ L[0,1] is bounded with w(t) ≥0 on [0,1] and w(t) >0 on [1/4,3/4],f ∈ C(R+,R+) with R+ = [0,∞), and D0+α is theαth Riemann- Liouville fractional derivative of h: [0,1]→R defined as
D0+α h(t) = 1 Γ(2−α)
d2 dt2
Z t 0
(t−s)1−αh(s)ds, whenever the right-hand side is defined.
Fractional differential equations have attracted extensive attention as they can be applied in various fields of science and engineering. Many phenomena in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc., can be modeled as fractional differential equations. We refer the reader to [10, 19] and references therein for the detail.
Due to the needs in applications, people have special interests in the existence of positive solutions of BVPs. In the study on integer order nonlinear BVPs, the fixed point theory on cones is a powerful tool in dealing with the existence of positive solutions. The main idea is to construct a cone in a Banach space and a
1991Mathematics Subject Classification. primary 34B15; secondary 34B18.
Key words and phrases. Positive solutions, fractional calculus, fixed point index.
†This author is supported by the NNSF of China (No. 10971231).
‡Corresponding author, min-wang@utc.edu.
completely continuous operator defined on this cone based on the corresponding Green’s function and then find fixed points of the operator. Many results have been obtained by this approach, see, for example, [5, 7, 8, 11, 14, 15, 16, 17, 18, 20, 21, 22]. This idea has also been used to the study of fractional BVPs, see [1, 2, 3, 6, 12, 13, 23, 25] and references therein for recent development.
In particular, Bai and L¨u [3] proved that function G : [0,1]×[0,1] → R+ defined by
G(t, s) =
[t(1−s)]α−1−(t−s)α−1
Γ(α) , 0≤s≤t ≤1, [t(1−s)]α−1
Γ(α) , 0≤t ≤s ≤1,
(1.3)
is the Green’s function for the BVP consisting of the fractional differential equation
D0+α u= 0
and BC (1.2). Based on it, they studied the BVP consisting of
−D0+α u=g(t, u), 0< t <1, (1.4) and BC (1.2) and obtained results on the existence of one or three positive solutions using the Krasnosel’skii fixed point theorem and the Leggett-Williams fixed point theorem. By further revealing the properties of the Green’s function G(t, s), Jiang and Yuan [12] obtained different conditions from those in [3] for BVP (1.4), (1.2) to have one positive solution and derived conditions for it to have two positive solutions.
To the best knowledge of the authors, there are no results yet on the existence of an arbitrary number of positive solutions of BVP (1.4), (1.2). This is due to the fact that although the Green’s function G(t, s) is positive in the interior of its domain {(t, s) : 0 ≤ t, s ≤ 1}, it becomes zero on the boundary, and does not satisfy the following condition
γΦ(s)≤G(t, s)≤Φ(s), t∈[a, b]⊂[0,1], s∈[0,1], (1.5) with a positive Φ andγ ∈(0,1). The lack of (1.5) prevents us from constructing a needed cone for the natural application of the fixed point theory as with positive Green’s functions.
In this paper, by carefully manipulating the Green’s function G(t, s), we obtain a weaker condition similar to (1.5) so that we are able to apply the fixed point theory on cones to BVP (1.1), (1.2), where the functions w(t) and f(x) satisfy certain conditions. A series of new criteria for BVP (1.1), (1.2) to have one, two, an arbitrary number, and even an infinite number of positive solutions are obtained. Our results reveal the fact that under our conditions, the existence of one or more positive solutions is determined by the behavior EJQTDE, 2012 No. 17, p. 2
of f on certain intervals. Moreover, a theorem on the nonexistence of positive solutions is also derived.
This paper is organized as follows: After this introduction, our main results are stated in Section 2. Several examples are given in Section 3. All the proofs of the main results are given in the last section.
2. Main results
The following assumptions are needed in the presentations of our main results.
(H1) f(x) > 0 on R+ and there exists a function B : (0,1) → (0,∞) such that for any r >0 and k∈(0,1)
maxx∈[0,r]f(x)
minx∈[kr,r]f(x) ≤B(k).
(H2) there exists k∗ ∈(0,1) such that
ψ(k∗) :=k∗(1 +B(k∗)M)≤γ; (2.1) whereM = max{M1, M2}with
M1 := max
s∈[1/4,1/2]
G(s−1/4, s−1/4)w(s−1/4)
G(s, s)w(s) and
M2 := max
s∈[1/2,3/4]
G(s+ 1/4, s+ 1/4)w(s+ 1/4)
G(s, s)w(s) ,
here Gis defined by (1.3); and γ = mins∈[1/4,3/4]γ(s) with
γ(s) =
[34(1−s)]α−1−(34 −s)α−1
[s(1−s)]α−1 , 0< s≤s1, 1
(4s)α−1, s1 ≤s <1,
(2.2)
here s1 is the unique solution of the equation [3
4(1−s)]α−1−(3
4 −s)α−1 = 1
4α−1(1−s)α−1. From the proof of [3, Lemma 2.4] we know that 1/4< s1 <3/4.
Remark 2.1. The Assumptions (H1) and (H2) are satisfied by a variety of functions of f and w.
(i) We claim that Assumption (H1) is satisfied by the following functions f1(x) =bxθ, b >0, θ >0,
f2(x) =b1xθ1 +b2xθ2, b1, b2 >0, 0< θ1 <1< θ2, and f3(x) =bx(sin(x) +µ), b >0, µ >1.
In fact, for any r >0 and k ∈(0,1), since maxx∈[0,r]f1(x)
minx∈[kr,r]f1(x) = brθ
b(kr)θ =k−θ, f1 satisfies (H1) with B(k) =k−θ; since
maxx∈[0,r]f2(x)
minx∈[kr,r]f2(x) = rθ2(b1rθ1−θ2+b2)
(kr)θ2(b1(kr)θ1−θ2 +b2) < k−θ2, f2 satisfies (H1) with B(k) =k−θ2; and since
maxx∈[0,r]f3(x)
minx∈[kr,r]f3(x) ≤ (µ+ 1)br
(µ−1)bkr = µ+ 1 (µ−1)k, f3 satisfies (H1) with B(k) = (µ+ 1)(µ−1)−1k−1.
(ii) It is easy to see that for any functionf satisfying Assumption (H1), As- sumption (H2) holds when the functionwis relatively small on [0,1/4]∪ [3/4,1] compared with its values on [1/4,3/4]. In this case, the number M in (H2) can be sufficiently small and hence inequality (2.1) has a solutionk∗ ∈(0,1).
However, the choice of the cut-off points 1/4 and 3/4 in the definitions of M1 and M2 is only for convenience in computations. Actually, all results in this paper can be extended to the case when the functionw(t) is sufficiently small near the endpoints 0 and 1.
(iii) For some functionsfsatisfying (H1), (H2) holds for anyw∈C([0,1],R+).
For example, consider f1(x) = bxθ with θ ∈ (0,1). Since B(k) = k−θ, ψ(k) = k+k1−θM → 0 as k → 0. Hence there exists k∗ ∈ (0,1) such that (2.1) holds.
In the following, we assume that the Assumptions (H1) and (H2) hold. Let L=γ
Z 3/4 1/4
G(s, s)w(s)ds, (2.3)
U = (1 +B(k∗)M) Z 3/4
1/4
G(s, s)w(s)ds. (2.4)
First, we state our basic result on the existence of a positive solution.
Theorem 2.1. Assume there existr∗, r∗ >0with[k∗r∗, r∗]∩[k∗r∗, r∗] =∅, such that
f(x)≤U−1r∗ on [k∗r∗, r∗], (2.5) and
f(x)≥L−1r∗ on [k∗r∗, r∗]. (2.6) EJQTDE, 2012 No. 17, p. 4
Then BVP (1.1), (1.2) has at least one positive solution u with min{r∗, r∗} ≤ kuk ≤max{r∗, r∗}.
In the sequel, we will use the following notation:
f0 = lim inf
x→0 f(x)/x, f∞= lim inf
x→∞ f(x)/x;
f0 = lim sup
x→0
f(x)/x, f∞= lim sup
x→∞
f(x)/x.
The next three theorems are derived from Theorem 2.1 using f0, f∞, f0, and f∞.
Theorem 2.2. BVP (1.1), (1.2) has at least one positive solution if either (a) f0 < U−1 and f∞>(k∗L)−1; or
(b) f0 >(k∗L)−1 and f∞ < U−1.
Theorem 2.3. Assume there exists r∗ >0 such that (2.5) holds.
(a) If f0 >(k∗L)−1, then BVP (1.1), (1.2) has at least one positive solution u with kuk ≤r∗;
(b) iff∞ >(k∗L)−1, then BVP (1.1), (1.2) has at least one positive solution u with kuk ≥r∗.
Theorem 2.4. Assume there exists r∗ >0 such that (2.6) holds.
(a) If f0 < U−1, then BVP (1.1), (1.2) has at least one positive solution u with kuk ≤r∗;
(b) if f∞ < U−1, then BVP (1.1), (1.2) has at least one positive solution u with kuk ≥r∗.
Combining Theorems 2.3 and 2.4 we obtain results on the existence of at least two positive solutions.
Theorem 2.5. Assume either
(a) f0 >(k∗L)−1 and f∞ >(k∗L)−1, and there exists r >0 such that f(x)< U−1r on [k∗r, r]; or (2.7) (b) f0 < U−1 and f∞< U−1, and there exists r >0 such that
f(x)> L−1r on [k∗r, r]. (2.8) Then BVP (1.1), (1.2) has at least two positive solutions u1 andu2 withku1k<
r <ku2k.
Note that in Theorem 2.5, the inequalities in (2.7) and (2.8) are strict and hence are different from those in (2.5) and (2.6) in Theorem 2.1. This is to guar- antee that the two solutions u1 and u2 are different. By applying Theorem 2.1 repeatedly, we can generalize the conclusion to obtain criteria for the existence of multiple positive solutions.
Theorem 2.6. Let {ri}Ni=1 ⊂ R such that 0 < r1 < r2 < r3 < · · · < rN and [k∗ri, ri], i= 1, . . . , N, are mutually disjoint. Assume either
(a) f satisfies (2.7)with r=ri wheni is odd, and satisfies (2.8)with r=ri
wheni is even; or
(b) f satisfies (2.7)withr =ri wheniis even, and satisfies (2.8)withr=ri wheni is odd.
Then BVP (1.1), (1.2) has at least N−1 positive solutions ui with ri <kuik<
ri+1, i= 1,2, . . . , N −1.
Theorem 2.7. Let {ri}∞i=1 ⊂R such that 0< r1 < r2 < r3 <· · · and [k∗ri, ri], i= 1, . . ., are mutually disjoint. Assume either
(a) f satisfies (2.5) with r∗ = ri when i is odd, and satisfies (2.6) with r∗ =ri when i is even; or
(b) f satisfies (2.5) with r∗ = ri when i is even, and satisfies (2.6) with r∗ =ri when i is odd.
Then BVP (1.1), (1.2) has an infinite number of positive solutions.
The following is an immediate consequence of Theorem 2.7.
Corollary 2.1. Let {ri}∞i=1 ⊂ R such that 0 < r1 < r2 < r3 < · · · and [k∗ri, ri], i = 1, . . ., are mutually disjoint. Let E1 = ∪∞i=1[k∗r2i−1, r2i−1] and E2 =∪∞i=1[k∗r2i, r2i]. Assume
lim sup
E1∋x→∞
f(x)
x < U−1 and lim inf
E2∋x→∞
f(x)
x >(k∗L)−1. Then BVP (1.1), (1.2) has an infinite number of positive solutions.
Our last theorem is about the nonexistence of positive solutions of BVP (1.1), (1.2).
Theorem 2.8. BVP (1.1), (1.2) has no positive solutions if f(x)/x < U−1 on (0,∞).
3. Examples
Example 1. Letf(x) = xθ with θ ∈(0,1). By Remark 2.1, Assumptions (H1) and (H2) are satisfied for any w ∈ C([0,1],R+). It is easy to see that f0 = ∞ and f∞ = 0. By Theorem 2.2 (b), BVP (1.1), (1.2) has at least one positive solution.
For easy computations, in Examples 2-4, we let α= 3/2 and w(t) =
c, 0≤t≤1/4,
1/G(t, t), 1/4< t <3/4,
c, 3/4≤t ≤1,
EJQTDE, 2012 No. 17, p. 6
where G(t, s) is given in (1.3) and c >0 is a constant. By some computations, we know that γ =√
3−2√
6/3 and M =√
3c/(4Γ(3/2)).
Example 2. Let f(x) = xθ with θ > 1. By a simple calculation we see that Assumptions (H1) and (H2) are satisfied when
c≤(4−8√
2/3)Γ(3/2)((θ−1)1/θ+ (θ−1)1−θθ)−θ.
In this case, since f0 = 0 and f∞ = ∞, by Theorem 2.2 (a), BVP (1.1), (1.2) has at least one positive solution.
Example 3. Letf(x) =λ(xθ1+xθ2), where 0< θ1 <1< θ2 <∞. By a simple calculation we see that Assumptions (H1) and (H2) are satisfied when
c≤(4−8√
2/3)Γ(3/2)((θ2−1)1/θ2 + (θ2−1)1−
θ2 θ2 )−θ2. Furthermore,L=√
3/2−√
6/3 andU = 1/2 +√
3c/(8Γ(3/2)kθ2). In this case, we let r1 = ((1−θ1)/(θ2−1))1/(θ2−θ1). We claim that BVP (1.1), (1.2) has
(a) at least one positive solution if λ= rθ11 +rθ12−1
U−1; (b) at least two positive solutions if λ∈(0, r1θ1 +r1θ2−1
U−1).
We note that f0 = f∞ =∞, f is strictly increasing, and r1 is the minimum point of f(x)/x on (0,∞). When λ = r1θ1 +r1θ2−1
U−1, f(x) ≤ U−1r1 on [k∗r1, r1]. By Theorem 2.3 (a), BVP (1.1), (1.2) has a positive solution u1 with ku1k ≤ r1. Similarly, by Theorem 2.3 (b), BVP (1.1), (1.2) has a positive solutionu2 with ku2k ≥r1. However, u1 and u2 may be the same when ku1k= ku2k=r1.
Part (b) follows similarly from Theorem 2.5 (a).
Example 4. Let L and U be defined by (2.3) and (2.4), and f(x) =
((2 +γ)x(sin(mlnx) +µ)/(2γL), x >0,
0, x= 0,
where
1< µ <min{4/(2 +γ),1 + 2γL/((2 +γ)U)} and
0< m <(π−2 sin−1δ)/ln(2/γ) with
δ= max{|δ1|,|δ2|},
hereδ1 = (4−2µ−µγ)/(2 +γ) and δ2 = 2γL/((2 +γ)U)−µ. Note that δ2 <0.
By a simple calculation we see that Assumptions (H1) and (H2) are satisfied for k∗ = γ/2 when c < γ16α−1(µ−1)Γ(α)31−α/(2(µ+ 1)). In this case, we claim that BVP (1.1), (1.2) has an infinite number of positive solutions.
In fact, it is easy to verify thatδ ∈(0,1). For i∈N, let ξi = exp(m−1(sin−1δ+ (i−1)π)), ηi = exp(m−1(iπ−sin−1δ)).
Then
ηi
ξi
= exp(m−1(π−2 sin−1δ))>exp(ln(2/γ)) =k∗−1. Hence ξi < k∗ηi. Then for any x∈[k∗ηi, ηi],x∈[ξi, ηi].
When i is odd, for any x∈[k∗ηi, ηi]
sin(mlnx)≥sin(mlnξi) = sin(sin−1δ) =δ > δ1. So
f(x)≥(2 +γ)k∗ηi(δ1+µ)/(2γL) =L−1ηi. When i is even, for any x∈[k∗ηi, ηi]
sin(mlnx)≤sin(mlnηi) = sin(−sin−1δ) =−δ≤δ2. So
f(x)≤(2 +γ)ηi(δ2 +µ)/(2γL) =U−1ηi.
Therefore, by Theorem 2.7 (b), BVP (1.1), (1.2) has an infinite number of positive solutions.
4. Proofs
LetX be a Banach space and K ⊂X a cone in X. Forr >0, define Kr ={u∈K| kuk< r} and ∂Kr ={u∈K| kuk=r}.
For an operator T : Kr → K, let i(T, Kr, K) be the fixed point index of T on Kr with respect to K. We will use the following well-known lemmas on fixed point index to prove our main results. For the detail, see [4, 9] and [24, page 529, A2, A3].
Lemma 4.1. Assume that for r > 0, T : Kr → K is a completely continuous operator such that T u6=u for u∈∂Kr.
(a) If kT uk ≥ kuk for u∈∂Kr, then i(T, Kr, K) = 0.
(b) If kT uk ≤ kuk for u∈∂Kr, then i(T, Kr, K) = 1.
Lemma 4.2. Let 0< r1 < r2 satisfy
i(T, Kr1, K) = 0 and i(T, Kr2, K) = 1;
or
i(T, Kr1, K) = 1 and i(T, Kr2, K) = 0.
Then T has a fixed point in Kr2 \Kr1.
EJQTDE, 2012 No. 17, p. 8
We will also use the following property of the Green’s function G(t, s) for BVP (1.1), (1.2), see [3, Lemma 2.4] for the proof.
Lemma 4.3. Let G(t, s) be defined by (1.3) and γ defined in (H2). Then (a) G(t, s)>0 on (0,1)×(0,1),
(b) max
t∈[0,1]G(t, s) =G(s, s) for 0< s <1 and min
1/4≤t≤3/4G(t, s)≥ γ G(s, s) for 1/4≤s≤3/4.
Let X =C[0,1] and kuk= maxt∈[0,1]|u(t)| for any u ∈X. Then (X,k · k) is a Banach space. Define
K ={u∈X |u(t)≥0 on [0,1] and min
t∈[1/4,3/4]u(t)≥k∗kuk}, (4.1) where k∗ is defined in (H2), and T :K →X as
(T u)(t) = Z 1
0
G(t, s)w(s)f(u(s))ds. (4.2)
Clearly, u(t) is a solution of BVP (1.1), (1.2) if and only if u ∈ K is a fixed point of T, andkT uk ≤R1
0 G(s, s)w(s)f(u(s))ds.
Lemma 4.4. Assume (H1), (H2) hold. Then T K ⊂K and T is a completely continuous operator.
Proof. For anyu∈K with kuk=r,k∗r ≤u(t)≤r on [1/4,3/4]. By (4.2) and Lemma 4.3, for t∈[1/4,3/4]
(T u)(t) = Z 1
0
G(t, s)w(s)f(u(s))ds
≥ γ Z 3/4
1/4
G(s, s)w(s)f(u(s))ds. (4.3)
By (H1), maxu∈[0,r]f(u)≤B(k∗) minu∈[k∗r,r]f(u).Then by (H2) Z 1/4
0
G(s, s)w(s)f(u(s))ds≤ max
u∈[0,r]f(u) Z 1/4
0
G(s, s)w(s)ds
≤ B(k∗) min
u∈[k∗r,r]f(u) Z 1/4
0
G(s, s)w(s)ds
≤ B(k∗)M1 min
u∈[k∗r,r]f(u) Z 1/2
1/4
G(s, s)w(s)ds
≤ B(k∗)M Z 1/2
1/4
G(s, s)w(s)f(u(s))ds. (4.4)
Similarly, Z 1
3/4
G(s, s)w(s)f(u(s))ds≤B(k∗)M Z 3/4
1/2
G(s, s)w(s)f(u(s))ds. (4.5) Therefore, by (4.4) and (4.5)
Z 1 0
G(s, s)w(s)f(u(s))ds
=
Z 1/4 0
+ Z 1/2
1/4
+ Z 3/4
1/2
+ Z 1
3/4
!
G(s, s)w(s)f(u(s))ds
≤(1 +B(k∗)M) Z 3/4
1/4
G(s, s)w(s)f(u(s))ds, (4.6) which follows that
Z 3/4 1/4
G(s, s)w(s)f(u(s))ds≥ 1 1 +B(k∗)M
Z 1 0
G(s, s)w(s)f(u(s))ds.
Hence by (H2) and (4.3)
(T u)(t)≥ γ 1 +B(k∗)M
Z 1 0
G(s, s)w(s)f(u(s))ds
≥ γ
1 +B(k∗)MkT uk ≥k∗kT ukon [1/4,3/4].
It is clear that (T u)(t) ≥ 0 on [0,1]. Therefore, T K ⊂ K. By Arzela-Ascoli
Theorem, T is completely continuous.
Proof of Theorem 2.1. Without loss of the generality, we assume r∗ < r∗. For any u ∈ ∂Kr∗, kuk = r∗ and k∗r∗ ≤ u(t) ≤ r∗ on [1/4,3/4]. By (2.4), (2.5), (4.2), (4.6), and Lemma 4.3 (a)
kT uk= max
t∈[0,1]
Z 1 0
G(t, s)w(s)f(u(s))ds
≤ Z 1
0
G(s, s)w(s)f(u(s))ds
≤ U−1r∗(1 +B(k∗)M) Z 3/4
1/4
G(s, s)w(s)ds=r∗. By Lemma 4.1 (a), i(T, Kr∗, K) = 1.
EJQTDE, 2012 No. 17, p. 10
For any u ∈ ∂Kr∗, kuk = r∗ and k∗r∗ ≤ u(t) ≤ r∗ on [1/4,3/4]. By (2.3), (2.6), (4.2), and Lemma 4.3 (b)
kT uk ≥(T u)(1/2) = Z 1
0
G(1/2, s)w(s)f(u(s))ds
≥ γ Z 3/4
1/4
G(s, s)w(s)f(u(s))ds
≥ L−1r∗γ Z 3/4
1/4
G(s, s)w(s)ds=r∗. By Lemma 4.1 (b), i(T, Kr∗, K) = 0.
By Lemma 4.2, T has a fixed pointu in K with r∗ ≤ kuk ≤r∗. Hence BVP (1.1), (1.2) has at least one positive solution u(t).
Proof of Theorem 2.2. (a) If f0 < U−1, there exists a sufficiently small r∗ >0 such that
f(x)< U−1x≤U−1r∗ on [k∗r∗, r∗], i.e., (2.5) holds.
If f∞ >(k∗L)−1, there exists ˆr > r∗ such that f(x)>(k∗L)−1x on [ˆr,∞).
Then for any r∗ with k∗r∗ ≥rˆ
f(x)>(k∗L)−1x≥L−1r∗ for all x∈[k∗r∗, r∗], i.e., (2.6) holds. Then the conclusion follows from Theorem 2.1.
(b) The proof is similar to Part (a) and hence is omitted.
The proofs of Theorems 2.3 and 2.4 are in the same way and hence are omitted.
Proof of Theorem 2.5. (a) If there exists r > 0 such that (2.7) holds, then by the continuity of f(x)/x on (0,∞), there exist r1, r2 >0 such that r1 < r < r2
andf(x)< U−1ri on [k∗ri, ri],i= 1,2. By Theorem 2.3 (a) and (b), BVP (1.1), (1.2) has two positive solutions u1 and u2 satisfying ku1k ≤r1 and ku2k ≥r2.
Similarly, Case (b) follows from Theorem 2.4.
The proofs of Theorems 2.6 and 2.7 are in the same way and are hence omitted.
Proof of Corollary 2.1. From the assumption we see that for sufficiently largei f(x)
x < U−1 on [k∗r2i−1, r2i−1]
and
f(x)
x >(k∗L)−1 on [k∗r2i, r2i].
This shows that for sufficiently large i
f(x)< U−1x≤U−1r2i−1 on [k∗r2i−1, r2i−1] and
f(x)>(k∗L)−1x≥L−1r2i on [k∗r2i, r2i].
Therefore, the conclusion follows from Theorem 2.7.
Proof of Theorem 2.8. Assume BVP (1.1), (1.2) has a positive solution u with kuk = r for some r > 0. Then u is a fixed point of the operator T defined by (4.2). For any t ∈[0,1], by (4.6)
u(t) = (T u)(t) = Z 1
0
G(t, s)w(s)f(u(s))ds
≤ (1 +B(k∗)M) Z 3/4
1/4
G(s, s)w(s)f(u(s))ds
< U−1r(1 +B(k∗)M) Z 3/4
1/4
G(s, s)w(s)ds=r,
which contradicts kuk = r. Therefore, BVP (1.1), (1.2) has no positive solu-
tions.
Acknowledgments
The authors are grateful to the referee for his/her careful checking and valu- able suggestions for changes which help us to improve the quality of the paper.
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(Received September 27, 2011)
Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA. Email: kong@math.niu.edu
Department of Mathematics, University of Tennessee at Chattanooga, Chat- tanooga, TN 37403, USA. Email: min-wang@utc.edu
